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The aim of the paper is to explore non-local reverse-space matrix non-linear Schrödinger equations and their inverse scattering transforms. Riemann–Hilbert problems are formulated to analyse the inverse scattering problems, and the Sokhotski–Plemelj formula is used to determine Gelfand–Levitan–Marchenko-type integral equations for generalised matrix Jost solutions. Soliton solutions are constructed through the reflectionless transforms associated with poles of the Riemann–Hilbert problems.
The purpose of this paper is to present an extension of the results in [8]. We establish a more general proof for the moving kernel formula to prove the spectral stability of periodic traveling wave solutions for the regularized Benjamin–Bona–Mahony type equations. As applications of our analysis, we show the spectral instability for the quintic Benjamin–Bona–Mahony equation and the spectral (orbital) stability for the regularized Benjamin–Ono equation.
We investigate, both analytically and numerically, dispersive fractalisation and quantisation of solutions to periodic linear and nonlinear Fermi–Pasta–Ulam–Tsingou systems. When subject to periodic boundary conditions and discontinuous initial conditions, e.g., a step function, both the linearised and nonlinear continuum models for FPUT exhibit fractal solution profiles at irrational times (as determined by the coefficients and the length of the interval) and quantised profiles (piecewise constant or perturbations thereof) at rational times. We observe a similar effect in the linearised FPUT chain at times t where these models have validity, namely t = O(h−2), where h is proportional to the intermass spacing or, equivalently, the reciprocal of the number of masses. For nonlinear periodic FPUT systems, our numerical results suggest a somewhat similar behaviour in the presence of small nonlinearities, which disappears as the nonlinear force increases in magnitude. However, these phenomena are manifested on very long time intervals, posing a severe challenge for numerical integration as the number of masses increases. Even with the high-order splitting methods used here, our numerical investigations are limited to nonlinear FPUT chains with a smaller number of masses than would be needed to resolve this question unambiguously.
We generalize Uhlenbeck’s generator theorem of ${\mathcal{L}}^{-}\operatorname{U}_{n}$ to the full rational loop group ${\mathcal{L}}^{-}\operatorname{GL}_{n}\mathbb{C}$ and its subgroups ${\mathcal{L}}^{-}\operatorname{GL}_{n}\mathbb{R}$, ${\mathcal{L}}^{-}\operatorname{U}_{p,q}$: they are all generated by just simple projective loops. Recall that Terng–Uhlenbeck studied the dressing actions of such projective loops as generalized Bäcklund transformations for integrable systems. Our result makes a nice supplement: any rational dressing is the composition of these Bäcklund transformations. This conclusion is surprising in the sense that Lie theory suggests the indispensable role of nilpotent loops in the case of noncompact reality conditions, and nilpotent dressings appear quite complicated and mysterious. The sacrifice is to introduce some extra fake singularities. So we also propose a set of generators if fake singularities are forbidden. A very geometric and physical construction of $\operatorname{U}_{p,q}$ is obtained as a by-product, generalizing the classical construction of unitary groups.
We consider the nonlinear wave equation (NLW) on the $d$-dimensional torus $\mathbb{T}^{d}$ with a smooth nonlinearity of order at least 2 at the origin. We prove that, for almost any mass, small and smooth solutions of high Sobolev indices are stable up to arbitrary long times with respect to the size of the initial data. To prove this result, we use a normal form transformation decomposing the dynamics into low and high frequencies with weak interactions. While the low part of the dynamics can be put under classical Birkhoff normal form, the high modes evolve according to a time-dependent linear Hamiltonian system. We then control the global dynamics by using polynomial growth estimates for high modes and the preservation of Sobolev norms for the low modes. Our general strategy applies to any semilinear Hamiltonian Partial Differential Equations (PDEs) whose linear frequencies satisfy a very general nonresonance condition. The (NLW) equation on $\mathbb{T}^{d}$ is a good example since the standard Birkhoff normal form applies only when $d=1$ while our strategy applies in any dimension.
Paraconformal or GL(2, ℝ) geometry on an n-dimensional manifold M is defined by a field of rational normal curves of degree n – 1 in the projectivised cotangent bundle ℙT*M. Such geometry is known to arise on solution spaces of ODEs with vanishing Wünschmann (Doubrov–Wilczynski) invariants. In this paper we discuss yet another natural source of GL(2, ℝ) structures, namely dispersionless integrable hierarchies of PDEs such as the dispersionless Kadomtsev–Petviashvili (dKP) hierarchy. In the latter context, GL(2, ℝ) structures coincide with the characteristic variety (principal symbol) of the hierarchy.
Dispersionless hierarchies provide explicit examples of particularly interesting classes of involutive GL(2, ℝ) structures studied in the literature. Thus, we obtain torsion-free GL(2, ℝ) structures of Bryant [5] that appeared in the context of exotic holonomy in dimension four, as well as totally geodesic GL(2, ℝ) structures of Krynski [33]. The latter possess a compatible affine connection (with torsion) and a two-parameter family of totally geodesic α-manifolds (coming from the dispersionless Lax equations), which makes them a natural generalisation of the Einstein–Weyl geometry.
Our main result states that involutive GL(2, ℝ) structures are governed by a dispersionless integrable system whose general local solution depends on 2n – 4 arbitrary functions of 3 variables. This establishes integrability of the system of Wünschmann conditions.
The unified transform method (UTM) or Fokas method for analyzing initial-boundary value (IBV) problems provides an important generalization of the inverse scattering transform (IST) method for analyzing initial value problems. In comparison with the IST, a major difficulty of the implementation of the UTM, in general, is the involvement of unknown boundary values. In this paper we analyze the IBV problem for the massive Thirring model in the quarter plane, assuming that the initial and boundary data belong to the Schwartz class. We show that for this integrable model, the UTM is as effective as the IST method: Riemann-Hilbert problems we formulated for such a problem have explicit (x, t)-dependence and depend only on the given initial and boundary values; they do not involve additional unknown boundary values.
Given a vector field on a manifold $M$, we define a globally conserved quantity to be a differential form whose Lie derivative is exact. Integrals of conserved quantities over suitable submanifolds are constant under time evolution, the Kelvin circulation theorem being a well-known special case. More generally, conserved quantities are well behaved under transgression to spaces of maps into $M$. We focus on the case of multisymplectic manifolds and Hamiltonian vector fields. Our main result is that in the presence of a Lie group of symmetries admitting a homotopy co-momentum map, one obtains a whole family of globally conserved quantities. This extends a classical result in symplectic geometry. We carry this out in a general setting, considering several variants of the notion of globally conserved quantity.
In this paper, we first construct π-type Fermions. According to these, we define π-type Boson–Fermion correspondence which is a generalization of the classical Boson–Fermion correspondence. We can obtain π-type symmetric functions Sλπ from the π-type Boson–Fermion correspondence, analogously to the way we get the Schur functions Sλ from the classical Boson–Fermion correspondence (which is the same thing as the Jacobi–Trudi formula). Then as a generalization of KP hierarchy, we construct the π-type KP hierarchy and obtain its tau functions.
We consider a broad class of systems of nonlinear integro-differential equations posed on the real line that arise as Euler–Lagrange equations to energies involving nonlinear nonlocal interactions. Although these equations are not readily cast as dynamical systems, we develop a calculus that yields a natural Hamiltonian formalism. In particular, we formulate Noether’s theorem in this context, identify a degenerate symplectic structure, and derive Hamiltonian differential equations on finite-dimensional center manifolds when those exist. Our formalism yields new natural conserved quantities. For Euler–Lagrange equations arising as traveling-wave equations in gradient flows, we identify Lyapunov functions. We provide several applications to pattern-forming systems including neural field and phase separation problems.
This paper introduces two novel conformal structure-preserving algorithms for solving the coupled damped nonlinear Schrödinger (CDNLS) system, which are based on the conformal multi-symplectic Hamiltonian formulation and its conformal conservation laws. The proposed algorithms can preserve corresponding conformal multi-symplectic conservation law and conformal momentum conservation law in any local time-space region, respectively. Moreover, it is further shown that the algorithms admit the conformal charge conservation law, and exactly preserve the dissipation rate of charge under appropriate boundary conditions. Numerical experiments are presented to demonstrate the conformal properties and effectiveness of the proposed algorithms during long-time numerical simulations and validate the analysis.
The Toda equation and its variants are studied in the filed of integrable systems. One particularly generalized time discretisation of the Toda equation is known as the discrete hungry Toda (dhToda) equation, which has two main variants referred to as the dhTodaI equation and dhTodaII equation. The dhToda equations have both been shown to be applicable to the computation of eigenvalues of totally nonnegative (TN) matrices, which are matrices without negative minors. The dhTodaI equation has been investigated with respect to the properties of integrable systems, but the dhTodaII equation has not. Explicit solutions using determinants and matrix representations called Lax pairs are often considered as symbolic properties of discrete integrable systems. In this paper, we clarify the determinant solution and Lax pair of the dhTodaII equation by focusing on an infinite sequence. We show that the resulting determinant solution firmly covers the general solution to the dhTodaII equation, and provide an asymptotic analysis of the general solution as discrete-time variable goes to infinity.
The KP-II equation was derived by Kadmotsev and Petviashvili to explain stability of line solitary waves of shallow water. Recently, Mizumachi proved nonlinear stability of 1-line solitons for exponentially localized perturbations. In this paper, we prove stability of 1-line solitons for perturbations in (1 + x2)−1/2−0H1(ℝ2) and perturbations in H1(ℝ2) ∩ ∂xL2(ℝ2).
Discrete linear Weingarten surfaces in space forms are characterized as special discrete $\unicode[STIX]{x1D6FA}$-nets, a discrete analogue of Demoulin’s $\unicode[STIX]{x1D6FA}$-surfaces. It is shown that the Lie-geometric deformation of $\unicode[STIX]{x1D6FA}$-nets descends to a Lawson transformation for discrete linear Weingarten surfaces, which coincides with the well-known Lawson correspondence in the constant mean curvature case.
In this paper, we propose a compact scheme to numerically study the coupled stochastic nonlinear Schrödinger equations. We prove that the compact scheme preserves the discrete stochastic multi-symplectic conservation law, discrete charge conservation law and discrete energy evolution law almost surely. Numerical experiments confirm well the theoretical analysis results. Furthermore, we present a detailed numerical investigation of the optical phenomena based on the compact scheme. By numerical experiments for various amplitudes of noise, we find that the noise accelerates the oscillation of the soliton and leads to the decay of the solution amplitudes with respect to time. In particular, if the noise is relatively strong, the soliton will be totally destroyed. Meanwhile, we observe that the phase shift is sensibly modified by the noise. Moreover, the numerical results present inelastic interaction which is different from the deterministic case.
We prove that a class of A-stable symplectic Runge–Kutta time semi-discretizations (including the Gauss–Legendre methods) applied to a class of semilinear Hamiltonian partial differential equations (PDEs) that are well posed on spaces of analytic functions with analytic initial data can be embedded into a modified Hamiltonian flow up to an exponentially small error. Consequently, such time semi-discretizations conserve the modified Hamiltonian up to an exponentially small error. The modified Hamiltonian is O(hp)-close to the original energy, where p is the order of the method and h is the time-step size. Examples of such systems are the semilinear wave equation, and the nonlinear Schrödinger equation with analytic nonlinearity and periodic boundary conditions. Standard Hamiltonian interpolation results do not apply here because of the occurrence of unbounded operators in the construction of the modified vector field. This loss of regularity in the construction can be taken care of by projecting the PDE to a subspace in which the operators occurring in the evolution equation are bounded, and by coupling the number of excited modes and the number of terms in the expansion of the modified vector field with the step size. This way we obtain exponential estimates of the form O(exp(–c/h1/(1+q))) with c > 0 and q ⩾ 0; for the semilinear wave equation, q = 1, and for the nonlinear Schrödinger equation, q = 2. We give an example which shows that analyticity of the initial data is necessary to obtain exponential estimates.
Integrable couplings of the Boiti-Pempinelli-Tu hierarchy are constructed by a class of non-semisimple block matrix loop algebras. Further, through using the variational identity theory, the Hamiltonian structures of those integrable couplings are obtained. The method can be applied to obtain other integrable hierarchies.
This paper explores the discrete singular convolution method for Hamiltonian PDEs. The differential matrices corresponding to two delta type kernels of the discrete singular convolution are presented analytically, which have the properties of high-order accuracy, bandlimited structure and thus can be excellent candidates for the spatial discretizations for Hamiltonian PDEs. Taking the nonlinear Schrödinger equation and the coupled Schrödinger equations for example, we construct two symplectic integrators combining this kind of differential matrices and appropriate symplectic time integrations, which both have been proved to satisfy the square conservation laws. Comprehensive numerical experiments including comparisons with the central finite difference method, the Fourier pseudospectral method, the wavelet collocation method are given to show the advantages of the new type of symplectic integrators.